Week 12
We had promised to teach you something profound that your professor does not know. Well, this week is the week. If your professor knows cycle formulas for dynamical averages, please send us her name, and we'll feature it in ChaosBook. They look like cumulants, but when you start to evaluate them you realize that you would not guess their form in 1007 Phys. Rev. Letters. Takes 12 weeks of hard study to start to understand them. 
 23. Cycle expansions
 24. Deterministic diffusion
 Homework 12
 Optional
Spectral eigenvalues and dynamical averages are computed by expanding spectral determinants into cycle expansions, expansions ordered by the topological lengths of periodic orbits. Skip sects. 23.3.1 "Newton algorithm for determining the evolution operator eigenvalues", and 23.6 "Stability ordering of cycle expansions".
Chapter 23 Cycle expansions  
Cycle expansions  slides  
Pseudocycles  
Evaluating a cycle expansion  
Probability conservation: An exact sum rule for cycles  
Leading eigenvalue of evolution operator yields dynamical averages  
Cycle formulas for dynamical averages  
Discrete vs continuous time averages  
Cycle formulas for finite grammar  
In real life, try stability ordering  
Summary, and what's next  
Talking Head video  Shadowing In this simulation Xiong Ding illustrates the basic principle of periodic orbit theory: the shadowing of a long periodic orbit by a pair of short ones: The theory works because long periodic orbits are shadowed by shorter ones. In this way the short periodic orbits build up accurate estimates of the infinite time behavior of a chaotic flow.  
Chapter 20 Averaging Read sect. 20.1, including sect. 20.1.3 "Moments, cumulants"  
Averages  a review  
Cumulants  
Expectation values  
Expectation values  from eigenvalue condition  
Cycle formula for expectation value  
?? 
We derive exact formulas for diffusion constants transport coefficients. All from first principles, without invoking any BoltzmannGibbs probabilistic notions. Read sect. 28.1: Foundations of statistical mechanics illuminated by 2dimensional Lorentz gas.
cycle expansion Due 19 April 2022 

Discussion forum for week 12 
How to best (and practically) evaluate cycle averaging formulas [Feb 2022 chat]  
Yasha Sinai gets Abel prize Sinai gets Abel Prize. Unexpectedly, Cvitanoviç nabs the coveted Night Fishing Champion trophy, to chagrin of the much more deserving Howey 3rd floor pros.  
Chaostheory pioneer Yakov Sinai nabs Abel Prize  
Sinai billiard [animation by Oliver Knill, 26 March 2014] 
We are grateful Doug Eardley, KITP, Santa Barbara CA, for recording the lecture 23 videos.